1) Yes you can download CorePlus to do that.
2) Possible with Same as 1. You can use core2usb
http://forum.tinycorelinux.net/index.php?topic=12235.03) Not possible
4) Yes this is the way Tiny Core works
5) Yes it would work. Unless kernel supports target laptop's hardware.
6) I did not see any script that install directly to HDD but its definitely possible.
7) Yes you can boot "Core" from HTTP any dot com without extensions. Possible via iPXE (i just tested booting not installing)
Its possible. You need setup tc-terminal-server
9) Simply yes.
10) I'm not an expert on this.
11) I don't know amazon's servers capabilities.
12) Yes, but old hardware may not simply boot.
13) iPXE supports some Wi-Fi hardware.
14) Actually it's not boots Tiny Core from Dropbox. It's emulated Tiny Core using QEMU
15) Yes. 56K modems can download data about at 50 kbit/s. It takes minimum about
30 minutes to download a 14 MB file.
16) Yes, but if only host system's console supports it